Evolution under multiallelic migration-selection models

The loss of a specified allele and the convergence of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection are investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Sufficient conditions are established for global fixation and for global loss of a particular allele. When migration is either sufficiently weak or sufficiently strong relative to selection, the equilibria are described, convergence of the gene frequencies is demonstrated, and sufficient conditions for the increase of a suitably defined mean fitness are offered. If the selection pattern is the same in every colony and such that in a panmictic population there is a globally asymptotically stable, internal (i.e., completely polymorphic) equilibrium point, then under certain weak assumptions on migration, the gene frequencies in the subdivided population converge globally to that equilibrium point. Thus, in this case, the ultimate state of the population is unaffected by geographical structure.     

Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.     

Patterns of multiallelic polymorphism maintained by migration and selection

Evolution at a multiallelic locus under the joint action of migration and viability selection is investigated. Generations are discrete and nonoverlapping. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. The forward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Several examples of globally attracting multiallelic equilibria are presented. Migration can cause global fixation even if, without migration, there is a globally attracting multiallelic equilibrium in every colony. Migration can also cause the global fixation of an allele that, without migration, is eliminated in every colony. Without dominance, generically, the number of alleles present at equilibrium cannot exceed the number of colonies. Some general properties and examples of the Levene model are studied in detail. If in each colony there is either no dominance or, without migration, a globally attracting internal equilibrium, then there exists a globally attracting equilibrium with migration. Therefore, if an internal equilibrium exists, it is the global attractor.     

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.     

Clines with partial panmixia in an unbounded unidimensional habitat

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection in an unbounded unidimensional habitat is investigated. Migration and selection are both weak. The former is homogenous and isotropic; the latter has no dominance. The population density is uniform. A simple, explicit formula is derived for the maximum value β₀ of the scaled panmictic rate β for which a cline exists. The former depends only on the asymptotic values of the scaled selection coefficient. If the two alleles have the same average selection coefficient, there exists a unique, globally asymptotically stable cline for every β≥0. Otherwise, if β≥β₀, the allele with the greater average selection coefficient is ultimately fixed. If β<β₀, there exists a unique, globally asymptotically stable cline, and some polymorphism is retained even infinitely far from its center. The gene frequencies at infinity are determined by a continuous-time, two-deme migration-selection model. An explicit expression is deduced for the monotone cline in a step-environment. These results differ fundamentally from those for the classical cline without panmixia.     

The number of heterozygous loci between two randomly chosen completely linked sequences of loci in two subdivided population models

The stationary probability distribution of the number of heterozygous loci in two randomly chosen sequences of completely linked infinite alleles loci, with mutation at each locus, is found in the island model for within and between islands. Results for an infinite site model are found as a limit. A single charge state locus is also studied in the island model and distributions found for the charge difference between two genes. Similar results are derived for a stepping stone model.     

Autosome STRs in native South America-Testing models of association with geography and language

Information on one Ecuadorian and three Peruvian Amerindian populations for 11 autosomal short tandem repeat (STR) loci is presented and incorporated in analyses that includes 26 other Native groups spread all over South America. Although in comparison with other studies we used a reduced number of markers, the number of populations included in our analyses is currently unmatched by any genome-wide dataset. The genetic polymorphisms indicate a clear division of the populations into three broad geographical areas: Andes, Amazonia, and the Southeast, which includes the Chaco and southern Brazil. The data also show good agreement with proposed hypotheses of splitting and dispersion of major language groups over the last 3,000 years. Therefore, relevant aspects of Native American history can be traced using as few as 11 STR autosomal markers coupled with a broad geographic distribution of sampled populations.     

Co-diversity and co-distribution in phyllostomid bats: Evaluating the relative roles of climate and niche conservatism

Explaining the causes of geographic gradients in biodiversity remains an elusive task. Traditionally, correlative approaches have been used to relate species richness with contemporary climate, without actually explaining the causal factors. Recent approaches propose simulation models as more appropriate tools for assessing potential causes of macroecological patterns. Here we developed stochastic models to assess the relative contribution of climate and niche conservatism in determining compositional similarity among sites (co-diversity) and geographic association among species (co-distribution) in the bat family Phyllostomidae. We used range-diversity plots and variance-ratio tests to describe and evaluate such patterns. Our results supported a strong effect of climate in determining cohesive ranges causing positive co-diversity and co-distribution. We also demonstrated a marginal effect of niche conservatism, as modeled here, among species in shaping these patterns. However, climate and niche conservatism are not sufficient and other processes are still required to explain observed patterns. Our study highlights the importance of historical processes and demonstrates the usefulness of a simulation framework in testing biogeographical hypothesis to understand the relationship between diversity and distribution.     

The number of equilibria in the diallelic Levene model with multiple demes

The Levene model is the simplest mathematical model to describe the evolution of gene frequencies in spatially subdivided populations. It provides insight into how locally varying selection promotes a population’s genetic diversity. Despite its simplicity, interesting problems have remained unsolved even in the diallelic case.In this paper we answer an open problem by establishing that for two alleles at one locus and J demes, up to 2J−1 polymorphic equilibria may coexist. We first present a proof for the case of stable monomorphisms and then show that the result also holds for protected alleles. These findings allow us to prove that any odd number (up to 2J−1) of equilibria is possible, before we extend the proof to even numbers. We conclude with some numerical results and show that for J>2, the proportion of parameter space affording this maximum is extremely small.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

Experimental demonstration of the impact of hard and soft selection regimes on polymorphism maintenance in spatially heterogeneous environments†

Predicting and managing contemporary adaption requires a proper understanding of the determinants of genetic variation. Spatial heterogeneity of the environment may stably maintain polymorphism when habitat contribution to the next generation can be considered independent of the degree of adaptation of local populations within habitats (i.e., under soft selection). In contrast, when habitats contribute proportionally to the mean fitness of the populations they host (hard selection), polymorphism is not expected to be maintained by selection. Although mathematically established decades ago, this prediction had never been demonstrated experimentally. Here, we provide an experimental test in which polymorphic populations of Escherichia coli growing in heterogeneous habitats were exposed to hard and soft selection regimes. As predicted by theory, polymorphism was preserved longer under soft selection. Complementary tests established that soft selection slowed fixation processes and could even protect polymorphism in the long term by providing a systematic advantage to rare genotypes.     

On the mutual maintenance of a polymorphism in two nonepistatic loci with partial selfing

In a two-locus system of multiplicative fitnesses with a mixed selfing and random mating, the maintenance of a polymorphism in each locus is not independent of its maintenance in the other one.     

Relationships between genotype x environment interaction and genetic correlation of the same trait measured in different environments

Summary Relationships between genotype x environment interaction and genetic correlation of the same trait measured in different fixed environments are derived by comparing the variance-covariance structures of observations between a one-way multiple-trait linear model and a two-way single-trait mixed linear model. In the latter model, heterogeneity of interaction variances among environments and non-zero covariances among interactions are assumed, in addition to the heterogeneity of error variances and non-zero covariances between genetic-group effects and interactions that were accommodated in earlier work. The results are applicable to more than two environments and to unbalanced data. This paper is a generalization and a correction of earlier works.     

Spatiotemporal environmental heterogeneity and the maintenance of the tailspot polymorphism in the variable platyfish (Xiphophorus variatus)

Genetic variation is critical for adaptive evolution. Despite its importance, there is still limited evidence in support of some prominent theoretical models explaining the maintenance of genetic polymorphism within populations. We examined 84 populations of Xiphophorus variatus, a livebearing fish with a genetic polymorphism associated with physiological performance, to test: (1) whether niche differentiation explains broad‐scale maintenance of polymorphism, (2) whether polymorphism is maintained among populations by local adaptation and migration, or (3) whether heterogeneity in explicit environmental variables could be linked to levels of polymorphism within populations. We found no evidence of climatic niche differentiation that could generate or maintain broad geographic variation in polymorphism. Subsequently, hierarchical partitioning of genetic richness and partial mantel tests revealed that 76% of the observed genetic richness was partitioned within populations with no effect of geographic distance on polymorphism. These results strongly suggest a lack of migration‐selection balance in the maintenance of polymorphism, and model selection confirmed a significant relationship between environmental heterogeneity and genetic richness within populations. Few studies have demonstrated such effects at this scale, and additional studies in other taxa should examine the generality of gene‐by‐environment interactions across populations to better understand the dynamics and scale of balancing selection.     

How to compute the effective size of spatiotemporally structured populations using separation of time scales

Calculations to derive effective population size become highly complicated when complex population structure is considered. We provide an easy method of computing the effective size of a subdivided population with overlapping generations (a spatiotemporally structured population) using an approximation based on separation of time scales. We also numerically compute the effective size to verify the accuracy of the derived formula. Various interesting quantities, including moments of coalescent time, are readily derived using this approach.     

Package models and the information crisis of prebiotic evolution

The coexistence between different types of templates has been the choice solution to the information crisis of prebiotic evolution, triggered by the finding that a single RNA-like template cannot carry enough information to code for any useful replicase. In principle, confining d distinct templates of length L in a package or protocell, whose survival depends on the coexistence of the templates it holds in, could resolve this crisis provided that d is made sufficiently large. Here we review the prototypical package model of Niesert et al. [1981. Origin of life between Scylla and Charybdis. J. Mol. Evol. 17, 348-353] which guarantees the greatest possible region of viability of the protocell population, and show that this model, and hence the entire package approach, does not resolve the information crisis. In particular, we show that the total information stored in a viable protocell (Ld) tends to a constant value that depends only on the spontaneous error rate per nucleotide of the template replication mechanism. As a result, an increase of d must be followed by a decrease of L, so that the net information gain is null.     

Shape and size variation on the wing of <Emphasis Type="Italic">Drosophila mediopunctata</Emphasis>: influence of chromosome inversions and genotype-environment interaction

The second chromosome of Drosophila mediopunctata is highly polymorphic for inversions. Previous work reported a significant interaction between these inversions and collecting date on wing size, suggesting the presence of genotype-environment interaction. We performed experiments in the laboratory to test for the joint effects of temperature and chromosome inversions on size and shape of the wing in D. mediopunctata. Size was measured as the centroid size, and shape was analyzed using the generalized least squares Procrustes superimposition followed by discriminant analysis and canonical variates analysis of partial warps and uniform components scores. Our findings show that wing size and shape are influenced by temperature, sex, and karyotype. We also found evidence suggestive of an interaction between the effects of karyotype and temperature on wing shape, indicating the existence of genotype-environment interaction for this trait in D. mediopunctata. In addition, the association between wing size and chromosome inversions is in agreement with previous results indicating that these inversions might be accumulating alleles adapted to different temperatures. However, no significant interaction between temperature and karyotype for size was found--in spite of the significant presence of temperature-genotype (cross) interaction. We suggest that other ecological factors--such as larval crowding--or seasonal variation of genetic content within inversions may explain the previous results.     

Combining ability analyses of stability parameters and forage yield in smooth bromegrass

Summary Twenty-one progenies of smooth bromegrass (Bromus inermis Leyss.) from a 7 X 7 half diallel cross, with their parents, were evaluated for three years at four locations in Alberta for the genetic variation of stability in expression of their annual yield. The linear response and deviation from linear response for each genotype were the two stability parameters considered, together with mean performance in the evaluation of each genotype. Four high yielding genotypes, namely 12, 13, 16, and 26, had general adaptability, while genotype 23 was especially suited to a poor environment. Combining ability analysis showed that general combining ability (GCA) and specific combining ability (SCA) were both important in the expression of yield. Inheritance of linear regression was controlled predominantly by GCA whereas both GCA and SCA were equally important in the expression of deviation. The presence of a substantial proportion of variability due to the additive genetic component in the linear response suggests that it should be possible to exploit this fraction of variability in developing high yielding stable cultivars.